3,042 research outputs found
The Igusa modular forms and ``the simplest'' Lorentzian Kac--Moody algebras
We find automorphic corrections for the Lorentzian Kac--Moody algebras with
the simplest generalized Cartan matrices of rank 3:
A_{1,0} =
2 0 -1
0 2 -2
-1 -2 2
and
A_{1,I} =
2 -2 -1
-2 2 -1
-1 -1 2
For A_{1,0} this correction is given by the Igusa Sp_4(Z)-modular form
\chi_{35} of weight 35, and for A_{1,I} by a Siege modular form of weight 30
with respect to a 2-congruence subgroup. We find infinite product or sum
expansions for these forms. Our method of construction of \chi_{35} leads to
the direct construction of Siegel modular forms by infinite product expansions,
whose divisors are the Humbert surfaces with fixed discriminants. Existence of
these forms was proved by van der Geer in 1982 using some geometrical
consideration. We announce a list of all hyperbolic symmetric generalized
Cartan matrices A of rank 3 such that A has elliptic or parabolic type, A has a
lattice Weyl vector, and A contains the affine submatrix \tilde{A}_1.Comment: 40 pages, no figures. AMS-Te
Scattering of circularly polarized light by a rotating black hole
We study scattering of polarized light by a rotating (Kerr) black hole of the
mass M and the angular momentum J. In order to keep trace of the polarization
dependence of photon trajectories one can use the following dimensionless
parameter: , where is the photon
frequency and the sign + (-) corresponds to the right (left) circular
polarization. We assume that |\varepsilonl << 1 and use the modified
geometric optics approximation developed in [1], that is we include the first
order in polarization dependent terms into the eikonal equation.
These corrections modify late time behavior of photons. We demonstrate that the
photon moves along a null curve, which in the limit becomes a
null geodesic. We focus on the scattering problem for polarized light. Namely,
we consider the following problems: (i) How does the photon bending angle
depend on its polarization; (ii) How does position of the image of a point-like
source depend on its polarization; (iii) How does the arrival time of photons
depend on their polarization. We perform the numerical calculations that
illustrate these effects for an extremely rotating black hole and discuss their
possible applications.Comment: 17 pages, 8 figure
A Toy Model for Topology Change Transitions: Role of Curvature Corrections
We consider properties of near-critical solutions describing a test static
axisymmetric D-dimensional brane interacting with a bulk N-dimensional black
hole (N>D). We focus our attention on the effects connected with curvature
corrections to the brane action. Namely, we demonstrate that the second order
phase transition in such a system is modified and becomes first order. We
discuss possible consequences of these results for merger transitions between
caged black holes and black strings.Comment: 11 pages, 9 figures, v2: published versio
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